Question

Determine the angle between the lines y=4x+2 and y=-x+3.

Answer 1

The vector $v_1=(4,1)$ is parallel to the line  $y=4x+2$
and the vector  $v_2(-1,1)$  is parallel to the line  $y=-x+3$.
On one hand  $v_1v_2=(-1)4+1^2=-3$.
On the other hand  
$v_1v_2=\vert v_1\vert\vert v_2\vert cos\theta\\=\sqrt{17}\sqrt{2}\cos\theta$

Equating we get  $\sqrt{34}\cos\theta=-3$.
so  $\theta=\cos^{-1}\dfrac{-3}{\sqrt{34}}=120\text{ degrees}$